# In the figure below, how do I find the value of the variable?

#### ns.19

##### New member
When I try to solve using the formula for the sum of the internal angles successively, I find a linear system with infinite solutions. Where am I wrong?

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#### tkhunny

##### Moderator
Staff member
Who says you are wrong? Have you yet used the fact that Triangle BDA is isosceles? That may lead somewhere.

#### ns.19

##### New member
Who says you are wrong? Have you yet used the fact that Triangle BDA is isosceles? That may lead somewhere.
I've tried, but, I couldn't solve this problem

#### ns.19

##### New member
Using the fact that the BDA triangle is isosceles and using the external angle theorem, can I conclude that the ADC angle measure is 80?

#### Jomo

##### Elite Member
If you show us your work then we can see if and where you are wrong? Please post your work as the forum guidelines request.

For the record I too get an infinite solution. I of course looked at BDA being isosceles but that did not help.
I recall a similar problem like this where Dr Peterson showed that there is a unique solution. If there is one I am sure that someone will point this out and state why. Meanwhile keep trying.

#### ns.19

##### New member
Right. First do: measure the angle ADC = d, measure the angle BDC = z and measure the angle BCD = y. Then, using the sum of the internal angles theorem in triangle ABC, we conclude that x + y = 50, using the same theorem in triangle BDC, we conclude that z + y = 110, using the same theorem in triangle ADC, d + x = 140. Finally, looking at the figure we have 160 + d + z = 360, that is d + z = 200. Setting up the linear system with this information, we can see that such a system has infinite solutions.

#### Jomo

##### Elite Member
Right. First do: measure the angle ADC = d, measure the angle BDC = z and measure the angle BCD = y. Then, using the sum of the internal angles theorem in triangle ABC, we conclude that x + y = 50, using the same theorem in triangle BDC, we conclude that z + y = 110, using the same theorem in triangle ADC, d + x = 140. Finally, looking at the figure we have 160 + d + z = 360, that is d + z = 200. Setting up the linear system with this information, we can see that such a system has infinite solutions.
Yes, that is just what I got, just with different lettering. I agree that the system you came up with does not have a solution. However what you have is more than just a system of equations, you have a geometric figure that can influence things. For example, d, z and y can't be negative. Possibly there is something else which we are missing that can make this problem uniquely solvable.

#### ns.19

##### New member
I understand, but, this question was passed on to me only in this way. Without any more information.

#### lev888

##### Full Member
Yes, that is just what I got, just with different lettering. I agree that the system you came up with does not have a solution. However what you have is more than just a system of equations, you have a geometric figure that can influence things. For example, d, z and y can't be negative. Possibly there is something else which we are missing that can make this problem uniquely solvable.
I think there is a unique solution (by construction?): ABC has angles 50, 50, 80. Point D is uniquely defined by the 10 degree angles. Therefore angle x is uniquely defined.
If you drop heights from D to AC and AB and make |AB| = 1 you can calculate everything using sin and cos of known angles. There is probably an easier way to do it.

#### Jomo

##### Elite Member
I understand, but, this question was passed on to me only in this way. Without any more information.
I am not saying that you did not supply all the information. I am saying that possibly we are missing something that is a result of what you gave us.

#### Jomo

##### Elite Member
I think there is a unique solution (by construction?): ABC has angles 50, 50, 80. Point D is uniquely defined by the 10 degree angles. Therefore angle x is uniquely defined.
If you drop heights from D to AC and AB and make |AB| = 1 you can calculate everything using sin and cos of known angles. There is probably an easier way to do it.
Nicely done! I am embarrassed at how poorly I do with geometry. I really have no talent in this subject. Your method was so obvious. Thanks for pointing it out to me and the OP and increasing my geometry maturity!